This is actually pretty simple to do, though I haven't found any English instructions for doing so as of this writing. I did, however, find a very informative page on Touhou music extraction which included instructions for SWR.

This tutorial is Windows-centric, since SWR is a Windows game to begin with. Even if you normally use a different operating system, you should have a method of running Windows and .NET applications available to you.

To begin with, you need:

• The list of music titles from Touhou SWR.
• This tool for extracting SWT .dat files.
• This command-line OGG decoder.
• The Touhou BGM Extractor.
• (Optional) LAME MP3 Encoder
• An installed copy of Scarlet Weather Rhapsody from which to extract the music!

If you have all of these things, then we're ready to begin extraction!

1. Extract oggdec.exe from the OGG decoder package and th105_bgm_tmp.bat from the music titles package into a directory somewhere on your hard drive. We'll call this the "working directory" for now.
2. Copy the file th105b.dat from your SWR game directory (wherever you installed it) to the working directory we created in step 1.
3. You should now be able to double-click and run the th105_bgm_tmp.bat file, which will use the OGG decoder and .dat file extractor to build a new temporary song data file.
4. Extract thbgmextractor to a directory of your choosing. This can be the working directory, or somewhere else.
5. Finally, extract the titles_th05.txt file from the music titles package into whichever directory you've extracted thbgmextractor to.

Now that you have a temporary song data file full of uncompressed music, you can use thbgmextractor to extract the music into standard PCM .wav files with titles. I've attached a rough translation of the thbgmextractor interface to this post. My Japanese isn't great, but it should give you a general idea. The tool is open-source, so I might release a translated version at some point, maybe.

You can re-compress these to MP3 format using a tool like LAME by selecting the option to do so inside thbgmextractor. If you just want the music and you don't care about titles or OGG compression, you can actually find the OGG music extracted directly from the game archive inside of the data/bgm subdirectory of the working directory.

Whatever you choose to do, you can delete the entire working directory once you've gotten the music you want from it.